Orbit Determination of the Lunar Reconnaissance Orbiter

Home , Apollo 12, Clementine (spacecraft), Icarus (crater), LCROSS, Lunar north pole, Lunar Prospector

JGeod DOI 10.1007/s00190-011-0509-4

ORIGINAL ARTICLE

Orbit determination of the Lunar Reconnaissance Orbiter

Erwan Mazarico · D. D. Rowlands · G. A. Neumann · D. E. Smith · M. H. Torrence · F. G. Lemoine · M. T. Zuber

Received: 17 March 2011 / Accepted: 13 August 2011 © Springer-Verlag 2011

Abstract We present the results on precision orbit deter- Keywords Moon · Orbit determination · Gravity · mination from the radio science investigation of the Lunar Altimetric crossover · Geodesy Reconnaissance Orbiter (LRO) spacecraft. We describe the data, modeling and methods used to achieve position knowledge several times better than the required 50–100 m (in total 1 Introduction position), over the period from 13 July 2009 to 31 January 2011. In addition to the near-continuous radiometric track- 1.1 Objectives ing data, we include altimetric data from the Lunar Orbiter Laser Altimeter (LOLA) in the form of crossover measure- The Lunar Reconnaissance Orbiter (LRO) mission (Chin ments, and show that they strongly improve the accuracy of et al. 2007) was designed to provide a comprehensive (global) the orbit reconstruction (total position overlap differences and detailed (high-resolution) survey of the Moon, in antic- decrease from ∼70 m to ∼23 m). To refine the spacecraft ipation of the needs of future orbital and landed exploration trajectory further, we develop a lunar gravity field by com- missions. The payload consists of seven instruments (Von- bining the newly acquired LRO data with the historical data. drak et al. 2010), some of which have very high spatial resolu- The reprocessing of the spacecraft trajectory with that model tion: better than 50 cm for the Lunar Reconnaissance Orbiter shows significantly increased accuracy (∼20 m with only the Camera (LROC) system (Robinson et al. 2010), ∼5-m foot- radiometric data, and ∼14 m with the addition of the altimet- prints and 10-cm vertical precision for the Lunar Orbiter ric crossovers). LOLA topographic maps and calibration data Laser Altimeter (LOLA, Smith et al. 2010a); 320 × 160 m from the Lunar Reconnaissance Orbiter Camera were used field-of-view for the Diviner Lunar Radiometer Experiment to supplement the results of the overlap analysis and demon- (DLRE, Paige et al. 2010); 30-m zoom resolution for the Min- strate the trajectory accuracy. iature Radio Frequency Technology Demonstration (Mini- RF, Nozette et al. 2010). The LRO datasets will ultimately be combined in a single reference frame based on the LOLA- Electronic supplementary material The online version of this defined geodetic grid, which requires accurate and consis- article (doi:10.1007/s00190-011-0509-4) contains supplementary tent trajectories for geolocation. As such, the LRO mission material, which is available to authorized users. defined a position knowledge requirement: 50–100 m total E. Mazarico · D. E. Smith · M. T. Zuber position and 1-m radially (Vondrak et al. 2010). Massachusetts Institute of Technology, Cambridge, USA Our goal is to report on the current status of the LRO precision orbit determination (POD). To determine precision E. Mazarico (B) · D. D. Rowlands · G. A. Neumann · D. E. Smith · M. H. Torrence · F. G. Lemoine orbits, the LRO trajectory from 13 July 2009 to 31 January NASA Goddard Space Flight Center, Planetary Geodynamics 2011 was reconstructed continuously, except for short gaps Laboratory, Greenbelt, USA near spacecraft maneuvers. As described below, the quality e-mail: erwan.m.mazarico@nasa.gov of the orbits is assessed primarily through overlap analysis, M. H. Torrence i.e., the consistency of trajectory segments when processed S.G.T. Inc., Greenbelt, USA in two consecutive time periods. 123 E. Mazarico et al.

To obtain the most precise orbits for LRO, we have devel- the spacecraft commissioning phase was initiated on 27 June oped a gravity model based on the historical tracking data 2009. The commissioning orbit was a quasi-frozen ∼30 × (Mazarico et al. 2010) and further tuned with data from LRO. 200 km polar orbit (i ∼ 90◦), with its periapsis near the lunar This intermediary model is not optimized for geophysical south pole. purposes and is intended to be used only to improve the LRO The instrument commissioning phase began on 3 July position knowledge. The upcoming Gravity Recovery and 2009. LOLA was initially turned on for 2 days at that time Interior Laboratory (GRAIL) mission (Zuber et al. 2011), and started continuous data collection on 13 July 2009. Since scheduled for launch in September 2011, should provide a then, it has operated continuously except for the monthly sta- lunar gravity field with far greater accuracy than what is pos- tion-keeping (SK) maneuvers, and a February 2010 safehold sible to achieve from analysis of LRO tracking data. event. On 15 September 2009, the spacecraft transitioned to its 1.2 Previous work nominal 2-h period mapping orbit, with lower eccentricity and an average altitude of 50 km. Because of gravitational In the recent era of lunar exploration, the Clementine space- perturbations, the orbit eccentricity and argument of periap- craft was the first to provide the data to determine the Moon’s sis naturally evolve, and monthly station-keeping maneuvers gravity field to relatively high spatial resolutions (l = 70, are necessary to preserve the mapping orbit (Houghton et al. Lemoine et al. 1997). The subsequent Lunar Prospector, 2007). The eccentricity is 0.0054 ± 0.0019, and never more with its lower orbit, in particular during the extended mis- than 0.010, with the spacecraft altitude generally between 35 sion (<50 km), enabled features as small as ∼40 km (l = and 65 km (referenced to a 1,737.4 km sphere). After 1 year 150−165) to be resolved (Konopliv et al. 2001). The recent of the nominal Exploration mission, dedicated to the survey JAXA SELENE mission consisted of three spacecraft: the of the polar regions and of 50 sites selected for NASA’s Con- main orbiter Kaguya, in a 100 km-altitude circular orbit, stellation human space flight program, the LRO mission was and two sub-satellites which participated in a VLBI exper- extended for 2 years by the NASA Science Mission Direc- iment. One of those sub-satellites also provided relay capa- torate and began its Science mission. bility for the radio signals, between ground stations and the In addition to the monthly SK maneuvers, smaller angular main orbiter. This enabled radiometric tracking of the far- momentum desaturation events (δH) generally occur twice side for the first time and led to great improvements in the a month: once just before an SK maneuver, and the other determination of the lunar farside gravity field (Namiki 2009; about 2 weeks later. When the Sun-orbit geometry is favor- Matsumoto et al. 2010; Goossens et al. 2011). able (large β angle1), enough power can be generated with the solar panel held fixed, which reduces the angular momen- 1.3 Overview tum accumulated in the reaction wheels and can allow a δH to be skipped (e.g., in July 2010). After a brief summary of the LRO mission as it relates to Other kinds of maneuvers are less frequent (yaw flip every the presented work, we describe the tracking data types used 6 months, at β ∼ 0◦ to keep the single solar panel oriented here: radiometric Doppler and range, and altimetric cross- towards the Sun), or unique (phasing maneuver to observe overs from LOLA. The orbit determination process is then the LCROSS impact in October 2009). Table S1 lists the described in detail, followed by a discussion of the various maneuvers of the nominal and science missions. steps in the current-best orbit reconstruction: radiometric- The orbital maneuver plan leaves LRO in free flight for only orbits, the addition of the altimeter data, and the inver- ∼14-day periods, which is desirable for orbit determination sion of a new LRO-tuned gravity field (LLGM-1, LRO Lunar and gravity field estimation. It enables orbit reconstruction Gravity Model), which is used to converge the final trajectory. to be executed over extended time spans, thereby facilitat- Finally, orbit overlap analyses are supplemented by results ing the recovery of long-period perturbations due to the long using independent data: LOLA topographic maps of the polar wavelengths of the gravity field (i.e., low-degree spherical regions and LROC calibration information. harmonic expansion coefficients, especially the zonal and resonant terms). It also improves the contribution of altimet- 2Data ric crossovers to the solution, by reducing the number of effectively independent spacecraft initial states that need to 2.1 Mission summary be estimated. The regularity of the spacecraft SK maneuvers leads to The LRO spacecraft was launched at 22:32 UTC on 18 June a natural partition of the mission into monthly periods, 2009 from Cape Canaveral Launch Complex 41, and entered lunar orbit on 23 June 2009. Five maneuvers designed to 1 Thebetaangle(β) is the viewing angle of the Sun from the orbit, gradually circularize the initial eccentric orbit followed, and i.e., the angle between the orbit plane and the spacecraft–Sun vector. 123 LRO orbit determination or “phases”, each lasting for approximately one lunation atic because they usually provide the strongest control of the (28 days). In the time period presented here (13 July 2009 spacecraft orbit (Table S2). Early in the mission, the Doppler to 31 January 2011), there are 21 phases in total. Commis- data collected by the White Sands station were time-biased by sioning, which lacks SK maneuvers, is divided into three 1/76th of a second (around −13.16 ms); this was corrected phases of equivalent duration (CO_01 to CO_03). The nom- on 19 October 2009. The Range data from all stations are inal mission is split into 13 phases (NO_01 to NO_13), and also affected by timing biases. Although these can be rather the currently ongoing science mission consists of 5 phases well characterized for each station (∼5 ms for White Sands, (SM_01 to SM_05), as of February 2011. Table 1 gives the and −2to−4 ms for USN stations), they are not constant per temporal extent of each phase. se, and still need to be systematically estimated. Figure S1 shows the range residuals from a sample arc (converged), 2.2 Data description with and without timing bias estimation.

2.2.1 Tracking data 2.2.2 Altimetric data

Radiometric tracking is the most widely used method of When available, accurate altimetric measurements can aug- tracking interplanetary spacecraft, providing good measurement the quality of the orbit reconstructions. At each ment accuracy and high operational availability. As for the intersection between two altimetry tracks (a “crossover” case of the Lunar Prospector (LP) spacecraft in 1998–1999, location), a constraint linking the height measurements made the LRO spacecraft is tracked continuously, i.e., anytime it at the two epochs and the surface height change can be cre- is visible from one of the ground network stations. However, ated. In the case of the Moon, with small-amplitude tides unlike LP which was tracked by the NASA Deep Space Net- (∼5–10 cm; Williams et al. 2008), no physical topographical work (DSN), LRO is tracked by a new LRO-dedicated NASA change above the instrument noise level (10cm) is expected, station in White Sands, New Mexico, and by the commercial so the reconstructed altitudes of the altimetric bounce points Universal Space Network (USN). In Table S2, we list the sta- should be identical. tion positions and their contributions to total tracking. All of Rowlands et al. (1999) used altimetric crossovers for the those stations produce S-band Doppler and range radiomet- orbit reconstruction of the Mars Global Surveyor (MGS) ric data, but only the White Sands station handles the science spacecraft and instrument pointing offset estimation of the data downlink (telemetry), at Ka-band frequencies. The data Mars Orbiter Laser Altimeter (MOLA; Zuber et al. 1992). are corrected for tropospheric delays from meteorological A limited amount of MOLA crossover data (∼21, 000 over data collected at the stations. a 25–30 day period) was used in the orbit determination The tracking strategy and S-band tracking precision of MGS and Mars gravity field inversion (Lemoine et al. requirements (1 mm s−1 for the White Sands station and 2001). A one-month simulation study of LRO (Rowlands 1.5–3 mm s−1 for the USN stations) were chosen because et al. 2009) indicated that significant improvements could be of the mission cost and the navigation positioning and pre- obtained from LOLA’s multi-beam configuration. LRO is the diction requirements (500-m total position), but were too first NASA lunar spacecraft to carry a laser altimeter with suf- large to enable the position reconstruction accuracy require- ficient capabilities to allow the use of altimetric crossovers in ments of the final LRO products (50-m total position). For the orbit determination process. Goossens et al. (2011) noted that reason, a Laser Ranging capability was added to the improvements in certain situations with the inclusion of al- spacecraft, providing one-way range measurements with timetric crossovers on the SELENE spacecraft, even though a precision of 5–10 cm averaged over 5 s (Zuber et al. the 1-Hz LALT instrument (Araki et al. 2009) has poorer 2010). However, the radiometric tracking performance is spatial resolution than the 28-Hz LOLA instrument. better than anticipated (∼0.3mms−1 and ∼0.2 m for White The Lunar Orbiter Laser Altimeter is a 10-cm-precision Sands, ∼0.4–0.8mms−1 and ∼0.4 m for USN; Table S2), 28-Hz, five-beam laser altimeter, described in detail by and greatly improves the achievable orbit accuracy. For this Smith et al. (2009). At the typical 50-km altitude, the five reason and because of the intricacies of the Laser Ranging 5-m-diameter footprints form a 25-m-radius cross, slanted by data (in particular, the estimation of station and spacecraft 26 degrees with respect to the direction of flight. With each clock and time bias parameters), early POD efforts and the laser shot, this pattern is repeated 56 m downtrack, and pro- results from the current work only include radiometric track- gressively produces five profiles, separated in the cross-track ing data. The use of Laser Ranging in LRO orbit reconstruc- direction by 10 m. Early results were presented by Smith et al. tion is however a subject of active research. (2010a). With its high pulse repetition rate and multi-beam The radiometric data are complicated by the presence of capability, the instrument collects up to 140 measurements biases which are typically not present in DSN tracking data. per second. As of 31 January 2011, the LOLA dataset con- The Doppler measurements are biased, which is problem- sists of 3.42 billion valid measurements. 123 E. Mazarico et al. 25 50 27 20 66 96 34 62 76 61 30 36 48 82 99 79 52 60 69 47 99 83 ...... Crossover RMS (m) LLGM-1 c overlap periods), number of LOLA 39 2 59 1 43 2 34 3 86 1 08 1 53 1 74 1 23 3 39 3 52 2 70 2 64 1 89 1 09 1 90 1 65 1 81 1 69 1 98 4 22 3 87 1 ...... Crossover RMS (m) GLGM-3 328 2 313 4 293 2 878 3 065 1 391 1 663 1 355 3 958 2 927 1 751 1 886 1 821 1 665 1 655 1 908 4 053 1 912 1 810 2 402 4 222 2 330 2 , , , , , , , , , , , , , , , , , , , , , , Number crossover locations 896 1 965 172 265 2 711 14 987 22 538 15 475 13 249 13 015 7 773 3 553 1 275 2 284 1 034 11 814 12 102 8 532 2 452 5 680 10 290 13 090 4 950 1 , , , , , , , , , , , , , , , , , , , , , , 176 Number (Range) , 928 159 559 164 055 3 373 154 595 159 943 161 037 134 936 169 549 150 273 163 697 142 755 154 098 152 411 132 009 151 324 158 204 149 742 159 192 130 290 145 230 145 910 137 , , , , , , , , , , , , , , , , , , , , , , 421 (Doppler) , 028 170 635 170 773 3 812 173 820 165 913 170 906 171 820 140 957 155 897 158 969 147 392 166 863 179 840 169 673 154 069 176 980 161 119 167 840 144 622 148 380 167 236 161 ...... For each orbital phase, we indicate: temporal extent and duration, number of radiometric data (Doppler and range; counted twice when falling in the ar NO_13 2010 AUG 20 15:25 2010 SEP 16 16:05 27 NO_12 2010 JUL 24 18:50 2 010 AUG 20 10:05 26 Total 2009 JUL 13 06:25 2011 JAN 31 18:40 560 NO_11 2010 JUN 27 20:12 2010 JUL 24 15:41 26 NO_05 2010 JAN 13 19:43 2010 FEB 09 15:24 26 NO_10 2010 MAY 30 19:20 2010 JUN 27 17:15 27 SM_05 2011 JAN 03 20:55 2011 JAN 31 18:40 27 NO_04 2009 DEC 17 21:09 2010 JAN 13 16:50 26 NO_03 2009 NOV 20 16:56 2009 DEC 17 15:55 26 NO_09 2010 MAY 03 18:52 2010 MAY 30 16:25 26 SM_04 2010 DEC 07 17:15 2011 JAN 03 16:30 26 SM_03 2010 NOV 11 02:15 2010 DEC 07 11:40 26 NO_02 2009 OCT 23 17:23 2009 NOV 20 14:05 27 NO_08 2010 APR 05 19:49 2010 MAY 03 15:59 27 NO_01 2009 SEP 26 22:20 2009 OCT 23 14:28 26 SM_02 2010 OCT 14 20:25 2010 NOV 11 12:15 27 NO_07 2010 MAR 09 17:23 2010 APR 05 16:54 26 CO_03 2009 AUG 31 12:20 2009 SEP 26 15:12 26 NO_06 2010 FEB 09 18:19 2010 MAR 09 14:29 27 SM_01 2010 SEP 21 00:00 2010 OCT 14 14:55 23 CO_02 2009 AUG 06 03:03 2009 AUG 31 12:10 25 ground track intersections between orbits occurring in that phases’ arcs, the crossover discrepancy RMS when converged with GLGM-3 or LLGM-1 Table 1 Phase StartCO_01 epoch 2009 JUL 13 06:25 Stop 2009 epoch AUG 06 12:05 24 Duration (days) Number

123 LRO orbit determination

a nation conditions with season. In periods of high β angle (|β|∼90◦), when the LRO orbit is nearly aligned with the terminator, the orbit-averaged lunar surface temperature is low, and the orbit-averaged return rate (percentage of good returns) drops dramatically (Fig. 1). However, the dependence on the β angle is very steep, and the instrument shows substantial instrument performance degradation only a few weeks per year. b The m-daily perturbations of the lunar geopotential (Kaula 1966) produce periodic perturbations in inclination; the most significant terms are the m = 1tom = 4, with periods of 28, 14, ∼10, and 7 days, respectively. Over the course of a month, the peak-to-peak variations are ∼1.3◦. These inclination changes, on an otherwise polar orbit, allow groundtracks to intersect and altimetric crossovers to be computed. Between 13 July 2009 and 31 January 2011, we found 5,007,658 crossover locations, potentially a slight overes- timate given our approach (valid data segments are picked Fig. 1 a Time series of the β angle (in degrees) between 13 July 2009 and 31 January 2011. For reference, the δH maneuvers are indicated by liberally to avoid missing any crossover). When the data a dashed line and the other maneuvers by a solid line. b Orbit-averaged are actually prepared, only crossovers with sufficient valid LOLA return rates (i.e., percentage of ground returns) for spot 1, spot 3 LOLA data are used to ensure the quality of the interpolation. and the 5 spots combined. Returns for spots 2 and 5 are similar to spot Figure 2a shows the distribution of those crossovers. It is 1; spot 4 follows spot 3 closely. The early lower return rates are due to the eccentric commissioning orbit, when LRO was flying high over the very heterogeneous, with extreme polar dominance, due to northern hemisphere. Later, the repeated dips are correlated with high the polar convergence of the LRO polar orbit: only 12, 5.7 β angle values, when the spacecraft is always near the terminator, and and 3% of the crossovers occur at latitudes less than 80◦, 60◦ are the result of the ‘LOLA anomaly’ and 40◦, respectively. The drape-like pattern is the result of the slow rotation of the Moon and the LRO high-inclination Compared to the MOLA data on Mars (Zuber et al. 1992; orbit, as noted by Rowlands et al. (2009), and the four longi- Smith et al. 2001), with its single-beam 160-m-diameter tude bands with fewer crossovers are due to the SK maneu- footprints spaced by ∼300 m, the higher resolution of the vers, which always occur at the same location, and lead to LOLA measurements improves the coverage around the the repeat of those monthly patterns. Figure 2b is identical groundtrack intersections, enabling more numerous (up to 25 to Fig. 2a, but with a resolution of 0.25◦ pixel−1 instead of effective crossovers at each groundtrack intersection because 2◦ pixel−1, which illustrates the fact that while saturating the of the five beams) and better-quality constraints through more polar regions, all the available LOLA crossovers so far only reliable interpolation. sparsely cover the equatorial regions. However, on the nightside, the data collection is ham- There are three main consequences of the LOLA anom- pered by a thermal alignment anomaly (Smith et al. 2010a). aly for the POD work presented here. First, the periods near When the instrument is exposed to cold temperatures, the β = 90◦ contain fewer crossovers because of the lower aver- thermal blanket contracts pulling the laser beam expander age return rate. This can affect the orbit reconstruction qual- out of alignment with the receiver telescope. The displace- ity of certain phases, in particular NO_03, NO_10, NO_11, ment is significant (∼600 µrad) and the laser spots move out and SM_05. In the future, this problem could potentially be of the detectors’ fields of view (∼400 µrad in diameter). For- mitigated by the use of multi-month crossover constraints tuitously, the displacement magnitude and direction (along (discussed in Sect. 3.3). the Y-axis in the LOLA frame) combine to pull the transmit- Second, the general strength of the crossovers is reduced. ted laser beams 2 and 5 into the fields of view 3 and 4, so that Because of the slow rotation of the Moon and the polar orbit altimetric data can still be collected the majority of the time. of LRO, the crossovers occur predominantly between ascend- Consequently, channels 3 and 4 show much better average ing and descending tracks. Given that we process crossover return rate than the others, as illustrated in Fig. 1. data in monthly batches, this means that in most intersecting Every orbit, altimetric data cannot be collected over short track pairs one will have been acquired on the dayside, and periods (typically less than a minute) around the termina- the other on the nightside. Thus, instead of the anticipated tor crossings, i.e., at the poles. Although the data density is geometry of five profiles intersecting five others (Rowlands reduced, these important regions can still be well mapped et al. 2009; Mazarico et al. 2010), we usually have two tracks thanks to the polar orbit convergence and changing illumi- intersecting five and only ten single-beam crossovers (the 123 E. Mazarico et al.

ac count 90 200

60 175

150 30 125 0 100

-30 75

-60 50

25 -90 0 30 60 90 120 150 180 210 240 270 300 330 360 0 0 30 60 90 120 150 180 210 240 270 300 330 360

b count d 90 10 9 60 8 30 7 6 0 5 -30 4 3 -60 2 -90 1 0 30 60 90 120 150 180 210 240 270 300 330 360 0 0306090 120 150 180 210 240 270 300 330 360

Fig. 2 Maps showing the counts of LOLA groundtrack intersections in actually used in the orbit determination process (i.e., only the cross- 2×2◦ (a)and0.25×0.25◦ (b) bins. The color scales focus on the lower overs occurring within each phase). The clipping is not as pronounced, counts to show the non-polar distribution. The pole values are strongly although the maxima are 1,276 (c) and 200 (d). The total count for (c) clipped, with actual count maxima of 17,929 (a) and 1,256 (b). The and (d) is 172,293. Frames (b)and(d) show that, although the coverage total count for (a)and(b) is 5,007,658. Frames (c)and(d)showsimi- is global, the possible sampling and actual sampling are still sparse lar maps, but of the counts of the LOLA crossover locations that were measurement type used here, similar to the MOLA cross- GEODYN was used in previous POD and gravity inver- overs used by Lemoine et al. 2001) are available, on average, sion work for the Moon (Lemoine et al. 1997; Mazarico et al. at each crossover location. 2010; Matsumoto et al. 2010; Goossens et al. 2011), Mars Third, for the same reason, much less “cross-track” infor- (Rowlands et al. 1999; Lemoine et al. 2001) and Mercury mation is available in one of the two intersecting tracks, (Smith et al. 2010b). Most of the supporting data, and force which significantly interferes with the recoverability of the and measurement models are similar to those described in relative geometric adjustment of both tracks. Only the (sin- previous studies. The lunar orientation and ephemeris are gle-beam) altimetric crossovers of the type used by Lemoine based on the JPL DE421 ephemeris (Williams et al. 2008). et al. (2001) with MGS/MOLA are employed, albeit more We used the degree 150 GLGM-3 gravity ﬁeld (Mazarico comprehensively than for MOLA and over a longer time et al. 2010) as a priori. That gravity solution is the result of period. the reprocessing at NASA GSFC of historical lunar tracking data (Lunar Orbiters, Apollo sub-satellites, Clementine and Lunar Prospector), in preparation for the LRO orbit deter- 3 Modeling and analysis mination effort. The purpose was to create normal equations of the existing data for integration with the upcoming LRO 3.1 Modeling data, as described in Sect. 3.3. The a priori value of the Love number k2 is 0.027 (following Lemoine et al. 1997 who used Precision orbit determination of the LRO spacecraft is per- the value determined from Lunar Laser Ranging by Williams formed with the GEODYN system of programs developed et al. 1987). The trajectory integration is performed with a at NASA Goddard Space Flight Center (Pavlis et al. 2006). 5-s time step. To compute the non-conservative accelerations, The spacecraft trajectory is integrated using a priori force the LRO spacecraft is modeled as a 10-plate macro-model models, with a number of model parameters estimated iter- and is oriented in space according to attitude telemetry in atively through batch least squares from the residuals of the the form of quaternions. The self-shadowed plate areas are observations with respect to the computed (modeled) values. computed as described in Mazarico et al. (2009). The lunar 123 LRO orbit determination surface albedo model is based on a low-degree spherical har- a monic expansion (Floberghagen et al. 1999). Planetary thermal radiation was not modeled, but its expected effect is small (a few meters, Mazarico et al. 2010) over the typical arc length. Following Mazarico et al. (2010), a constant along-track empirical acceleration is estimated to reduce the along-track orbit reconstruction error; the adjusted accelerations are small, around −2 ± 6 × 10−10 ms−2. The center- b of-mass position with respect to the spacecraft body frame is considered variable and is computed based on the fuel mass and the orientation of the solar panel and high-gain antenna (HGA). Because the two gimbals of the LRO HGA orientation system are separated by ∼10 cm (which is the LOLA-LR instrument precision), the HGA phase center offset is modeled by two non-juxtaposed single-axis gimbals. The laser altimeter data are modeled from the LOLA receiver telescope position, and constant roll and pitch pointing biases around Fig. 3 Doppler (a) and range (b) residuals are plotted against the cosine the post-calibration boresight vector are estimated. The data of the α angle (Earth viewing angle, i.e., elevation of the Earth from the weights were set to values several times above the intrinsic orbit plane). Face-on or edge-on (as seen from the Earth) geometries noise and residual root mean square (RMS): 1 mm s−1 for plot to the left or right, respectively. The residuals obtained after con- the radiometric Doppler, 10 m for the radiometric range, and vergence with two a priori gravity ﬁelds (GLGM-3 and LLGM-1) are shown 1 m for the altimetric crossovers.

3.2 Initial data analysis navigation team (Flight Dynamics Facility, at NASA GSFC) daily reconstructions. After a one-iteration run to manually 3.2.1 Radiometric-only orbits edit out the anomalous Doppler and Range measurements, the arc is iteratively converged, by jointly estimating the We process the data in several stages because the measure- values of typical arc parameters (initial state, measurement ment types require different levels of processing effort: the biases, radiation pressure scale factor and empirical constant radiometric data can be used readily and independently, but along-track acceleration). the altimetric crossovers, although they strengthen the solu- The data reduction is generally of good quality, with tion, cannot stand on their own. Doppler and range RMS values between 0.4 and 1.5mms−1 The radiometric data are divided into time periods called and between 1 and 4 m, respectively (Fig. 3). The arc RMS ‘arcs’, to be integrated and converged individually. Follow- values are strongly correlated with the Earth viewing geom- ing previous work on Lunar Prospector (Konopliv et al. 2001; etry angle α2: lower for face-on geometries (cos α = 0) and Mazarico et al. 2010), we used short arcs, with an aver- higher for edge-on (cos α = 1, when the orbit crosses the age duration of 2.5 days. This length allows the arc to start deep far side). The better ﬁts do not indicate better orbits: and end with tracking passes from the higher-quality White when face-on, the line-of-sight direction is aligned with the Sands station. Although not necessary, this 8- to 12-h over- horizontal component of the gravitational residual acceler- lap period proves an important and consistent monitor of ation and expected to be smaller than the radial component the orbit reconstruction quality, as discussed below. These observed in edge-on orbits. short arcs are adapted to high-quality orbits: long enough We can assess the orbit precision (repeatability) by evalu- to include three whole White Sands tracking sessions and a ating the RMS position difference of the trajectories com- number of intervening USN tracking passes, but short enough puted in consecutive arc pairs during their overlapping to avoid the detrimental build-up of modeling-related errors. period. This is done in the along-track (A), cross-track (C) In total, we constructed 272 short arcs to fully cover the 21 and radial (R) directions (ACR), as well as in terms of total analyzed phases. Figure S2 shows a typical arc distribution position (T). Over the whole period, there are 224 overlaps. over 1 month. (Some potential overlapping periods are voided by maneu- We process the data in monthly batches, well after they vers.) The overlap duration is 10.5 ± 2 h (see Figure S3 for have been collected. Each phase is divided into 12 to 14 short the distribution histogram). arcs, which are individually cleaned and converged. In this ﬁrst step, we only use the radiometric data. The initial state 2 The alpha angle (α) is the viewing angle of the Earth from the orbit, (position and velocity) of the spacecraft is obtained from the i.e., the angle between the orbit plane and the spacecraft–Earth vector. 123 E. Mazarico et al.

Fig. 4 Phase-average overlap RMS values in the along-track a (a), cross-track (b) and radial (c) directions and in total position (d) for various gravity ﬁelds and data usage. Blue indicates GLGM-3 radiometric-only orbits; cyan GLGM-3 with radiometric and altimetric data. Similarly, LLGM-1 radiometric-only orbits are shown in red, and the orbits with additional altimetric crossovers b in magenta

The quality of the orbits reconstructed with the radiomet- is a pre-requisite for the use of altimetric crossovers. Indeed, ric data alone is close to the LRO requirements, although given the slow rotation of the Moon and the polar inclination some periods can show larger discrepancies. In Fig. 4,we of LRO, altimetric crossovers mostly occur between tracks show the “phase”-averaged overlap RMS time series. Table 3 ∼14 days apart (or any multiple). gives an overall summary of the overlap differences: 47.6m One month of individual short arcs are processed simulta- along-track, 49.9m cross-track, 4.5m radial, and 70.1m total neously by GEODYN, similar to the analysis of GPS data position. The overlap RMS values calculated over each phase (Luthcke et al. 2003). Altimetric crossovers provide the are presented in Table S3. In general, along-track and cross- “connection” between the separate data arcs. The ∼28-day track overlaps are smaller in edge-on and face-on geometries, duration is chosen to allow any orbit to share an altimet- respectively (Figure S4). ric crossover with another orbit, which would not be the case with a 14-day duration: only the very ﬁrst and very last orbits would have crossovers. Crossover computations inevitably 3.2.2 Addition of altimetric crossovers require non-sequential accessing of orbital information. On each iteration, the numerical integration of orbits, and asso- Once all the arcs of a given phase have been converged with ciated partial derivatives, is carried out for the whole arc in the radiometric data, they are ready to be combined, which advance of crossover computations, and only certain quan-

123 LRO orbit determination

Table 2 Comparison of the values obtained by various studies for key low-degree gravitational coefﬁcients: GM (a), offset by −4.9028 × 12 3 −2 −5 10 m s ; C20 (b), offset by +9.08938 × 10 ; C21 (c); S21 (d); and k20 (e) 9 9 Reference GM − 4.9028 × (C20 +9.08938× C21 × 10 S21 × 10 k20 1012 (m3 s−2) 10−5) × 109

Dickey et al. (1994)– −−−0.03020 ± 0.0012 Williams et al. (2009)– −−−0.0199 ± 0.0025 Williams et al. (2011)– −−−0.0229 ± 0.0020 LP100K 238,000 ± 20,600 20.009 ± 5.319 8.383 ± 4.292 7.707 ± 0.350 n/a LP150Q 1,076,100 ± 8,100 −7.295 ± 5.348 −1.863 ± 4.409 −1.425 ± 0.335 0.0248 ± 0.0030 Goosens and Matsumoto 2008 – −−−0.0244 ± 0.0080 SGM100H 2,121,666 ± 13,000 −91.596 ± 1.631 11.270 ± 1.064 −15.894 ± 1.443 0.0240 ± 0.0015 SGM100I 225,153 −102.97 ± 1.650 7.861 ± 1.045 −10.809 ± 1.349 0.0255 ± 0.0016 SGM150 795,109 6.832 ± 2.463 28.776 ± 1.534 8.248 ± 1.547 n/a GLGM-3 238,000 (not adjusted, 3.654 ± 5.344 −29.620 ± 7.719 27.715 ±−4.357 0.0235 ± 0.0020 =LP100K) LLGM-1 105,594 ± 9,980 5.102 ± 1.916 −52.865 ± 1.330 −13.345 ± 1.570 0.0270 ± 0.0004

References are given in the text. The uncertainties quoted are the 1-sigma values, except for k20 where we report ten times the spacecraft-derived formal errors following Konopliv et al. (2001)andMatsumoto et al. (2010) tities from the integration are stored in memory. This way, Out of those ∼172, 000 crossover locations, about ∼159, 000 a balance is struck between memory requirements and the are retained by GEODYN and contribute to the trajectory computational requirements associated with non-sequential adjustment. In terms of actual measurements used, ∼951,000 accessing of orbital information. (single-beam) crossovers exist at those ∼172,000 locations, Because of the LOLA anomaly, we know that the laser of which ∼37,000 are rejected during convergence. Editing altimeter alignment changes significantly between the day- criteria are high RMS of polynomial fit to the topography, side and the nightside, and that it varies rapidly near the high-slope terrain, large altitude discrepancy between the terminator. However, little data are collected during those two tracks and sensitivity with respect to the arc parameters short transition times (Sect. 2.2.2), which makes it difficult (i.e., large state partial derivatives). The dominant factor for to characterize the alignment other than by constant offsets LRO is the latter, with some crossovers inclined to disturb from the nominally calibrated pointing. Thus, two indepen- the trajectory too much compared to what we expect from dent sets of altimeter pointing biases (in roll and pitch) are the majority. Compared to the MOLA analysis (Lemoine estimated: one for the five daytime spots, and another for the et al. 2001), the maximum slope was relaxed (from 0.25◦ two nighttime spots. Because of the large disparity in cross- to 1.0◦), thanks to the shorter along-track distance between over temporal density, the adjusted values are really reliable consecutive bounce points and the resulting more reliable only when estimated once a month (i.e., the same adjustment interpolation. applied to all the arcs). After orbit adjustment, the remaining height differences With the multi-satellite approach outlined above, we are (crossover discrepancies) are small and comparable to the restricted to a narrow subset of the ∼5 million LOLA cross- radial direction overlap (Tables 1, 2). The overall residual over locations at hand (Sect. 2.2.2). Because we process data RMS is ∼2.4 m. As noted above, some months, in particular in monthly batches, only the crossovers occurring within each at high β angles, contain fewer crossovers than nominally month (“intra-month crossovers”) are available. Multi-month possible. In those cases, the crossover measurements do not arcs, designed to access the more numerous “inter-month significantly help the orbital solution. Figure 5 shows the crossovers”, are possible but computationally challenging distributions of both edited and retained crossover discrepan- and of limited interest to orbit reconstruction work due to cies, for a strong phase (NO_01) and a weak phase (NO_03). the diminishing return. Figure 2c, d shows the spatial dis- Only ∼12% of the crossovers are rejected in the first case, tribution of the crossovers actually used. For each phase, but ∼49% of an already smaller number are discarded for we choose the crossovers which have enough LOLA data NO_03. Nevertheless, the RMS discrepancy does not exceed (for interpolation quality) and yield good adjustments, and 5 m (cf. Table 1). As shown in Table S3, the phase NO_03 then select the appropriate individual range measurements. deteriorated in along-track and radial consistency, and the Table 1 indicates the number of crossover locations available crossovers provided only marginal total position improve- within each phase, with a strong β dependence (Sect. 2.2.2). ments. This is in contrast with the spectacular enhancement

123 E. Mazarico et al.

re-weighting of the various measurement types during the inversion). For practical reasons, we worked only on phases CO_01 to NO_13. Substantial changes were made to GEODYN to enable the computation of the normal equations for the crossover measurements. We mentioned above the need to pre-integrate the trajectory to use these data for orbit convergence (i.e., store the partial derivatives of the initial state and other arc parameters), but this need is exacerbated by the large number of gravitational Stokes coefficients to be estimated (22,797 for a degree-150 gravity field). We use a multi-step approach, with large temporary files to store the necessary information. In addition, while the overlapping short arcs are important in the assessment of the orbit quality, it would be misguided to have some of the data (every other White Sands pass) counted twice in the normal equations. Instead of just trun- ∼ Fig. 5 Histogram of the crossover discrepancy (interpolated altitude cating most of the short arcs by 12 h (to remove the last difference at point of intersection of two LOLA spot groundtracks) for White Sands pass, for example), which would make them two different multi-satellite arcs: phase NO_01 (2009-09-26 to 2009- weaker, we created an alternate set of 91 longer ∼5-day arcs 10-23, in blue) and phase NO_03 (2009-11-20 to 2009-12-17, in red). (compared to 207 short arcs over the same time period). The In each case, the solid line shows the distribution for the (single-beam) crossovers retained in the solution and the dashed line that of the rejected quality of those arcs can of course not be assessed directly crossovers. While the strong NO_01 month has only ∼12% of its numer- through overlap analysis, but orbit differences with the bet- ous crossover measurements excluded, the NO_03 month, with already ter-characterized short arcs are small. The average RMS orbit scarcer crossovers, is further weakened by the rejection of nearly half differences are 26.5, 13.5, 2.8 m in the ACR directions, and the crossover data. Note the logarithmic scale for the horizontal axis 32.8 m in total position, i.e., smaller (about half) than the short arc overlap values, giving us confidence that the longer observed over most orbital phases (Table S3): the overall arcs are appropriate for building normal equations. overlap RMS values decrease from ∼70 m in total position Normal equations of the historical lunar radiometric track- to ∼23 m (Table 3), a 70% reduction. ing data, recently analyzed for the creation of the GLGM-3 Obviously then, the use of altimetric crossovers in the gravity field (Mazarico et al. 2010), were recomputed with orbit determination process can help meet the LRO position GLGM-3 as an a priori. They were then combined with the knowledge requirements, also showing that they can substan- LRO normal equations and the gravity field inversion was tially mitigate the gravity-related modeling errors. performed. Even though our a priori weighting of the radiometric data was appropriate, the altimetric crossover data had 3.3 Gravity field inversion to be downweighted, because of the large-amplitude oscilla- tions they induced in the farside gravity field, even with the The natural next step is to use the signals still present in presence of a Kaula constraint. We also needed to discard the measurement residuals to improve the gravitational force the crossover normal equations of a few phases, before the modeling. We use GEODYN to create one set of normal obtained farside gravity field did not show excessive power equations per phase and per measurement type (i.e., Dopp- (compared to GLGM-3 and SGM100i from Goossens et al. ler, range and altimetric crossover separately, which allows 2011). While we continue to tackle this issue, a possible

Table 3 Average overlap RMS differences, in meters, in the along-track (A), cross-track (C) and radial (R) directions, as well as in total position (T) A priori gravity Radio−only orbit overlaps (m) Radio+crossover orbit overlaps (m)

Along Cross Radial Total Along Cross Radial Total

GLGM-3 all phases 47.57 49.91 4.50 70.06 17.22 14.26 3.96 22.91 GLGM-3 NO_01 to NO_13 41.81 40.87 3.47 58.77 16.38 13.72 3.66 21.82 LLGM-1 all phases 16.71 15.44 1.58 23.39 10.18 8.37 1.80 13.63 LLGM-1 NO_01 to NO_13 10.67 10.09 1.13 14.79 8.35 8.21 1.62 12.00 For each a priori gravity ﬁeld (GLGM-3 or LLGM-1), the averages are calculated over all 21 phases and over the 13 nominal phases. Results with and without the altimetric crossovers are given. More details (phase averages) are given in Tables S2 and S3

123 LRO orbit determination explanation is the sparsity of the crossover data of the farside C41 are rather large (0.64, −0.79 and −0.65, respectively). (Fig. 2c-d): some data can outweigh the Kaula constraint, but They are, however, consistently smaller than in the GLGM-3 are too few to yield robust estimates of the spherical harmon- solution, likely due to the addition of the LRO data. Our k2 ics expansion coefficients. value is very close to our a priori value based on Williams Our selected gravity field, LLGM-1 (LRO Lunar Gravity et al. (1987). Other groups’ estimates are consistently lower, Model), is not manifestly superior than GLGM-3 in terms of in the 0.023–0.026 range, as is our GLGM-3 adjustment geophysical performance, based on admittance and correla- (0.0235 ± 0.0002). We note however the latest SELENE tion studies (following Mazarico et al. 2010; not presented estimate using VLBI (Goossens et al. 2011) data is clos- here). Coefficient differences between LLGM-1 and GLGM- est to our number (0.0255 ± 0.00016). Estimates based on 3 are shown in Figure S5. The low-order coefficients at high Lunar Laser Ranging, after a long decrease due to model- degree show stronger changes compared to other high-degree ing change over the past decade (0.0302 by Dickey et al. coefficients. This is consistent with the LRO data affect- 1994 to 0.0199 by Williams et al. 2009), have recently been ing the determination of low-order coefficients through the revised to values more consistent with satellite-based results m-daily perturbations. The relatively large coefficient differ- (0.0229 ± 0.0020 by Williams et al. 2011). The correlation ences in the l = 5–70 band are due to a combination of the of our k2 estimate with the other low-degree coefficients is Kaula-type decrease in coefficient power with degree, and to small, with a maximum (absolute) value near 0.27 with S30, the re-arrangement of the lumped coefficients, affected by the followed by S21, C22 and S22 (smaller than 0.25). This is an farside gravity field. When expanding both the GLGM-3 and improvement over GLGM-3, where the maximum correla- LLGM-1 fields, we find that the gravity anomaly changes are tion was 0.3, with S21. small on the nearside (<10 mgal RMS excluding the polar regions) and larger on the farside, as expected (∼62 mgal 3.4 Precise orbit reconstruction RMS). We did expect a better determination of the farside gravity anomalies thanks to the altimetric crossovers. How- To further improve the orbit accuracy, we reprocessed the ever, in the non-polar regions, as mentioned above, the cov- LRO tracking data, by using the new LLGM-1 gravity field erage by LOLA crossovers actually used in the solution is as an a priori to reprocess the short arcs. The radiometric- very sparse. In the polar regions (down to ∼70◦ latitude), only arcs show improved residual Doppler and range RMS, the lunar farside gravity field was already well character- with maxima around 1 mm s−1 and 4 m, respectively. This ized by the historical tracking data. However, due the great is mainly due to a better fit when the orbit is seen edge-on majority of LOLA crossovers occurring in that region, we (Fig. 3), presumably because of a better (LRO-tuned) mod- see larger changes than on the nearside (∼42 mgal RMS). eling of the integrated farside gravitational effects. Those are appreciable, although significantly smaller than The radiometric-only orbit performance is also excellent, the ∼83 mgal RMS observed in the deep farside (spherical with dramatic improvements compared to GLGM-3 in all cap of half-angle 60◦, centered on 180◦E, 0◦N). Multi-month directions (see Fig. 4 and Table S4). The mission-averaged arcs can allow us to access the complete set and potentially total position knowledge is around ∼23 m RMS (Table 3), at improve the farside gravity anomalies. This computationally the level obtained with altimetric crossovers using GLGM-3 intensive approach may be the subject of future work. Our as the a priori model. Moreover, we note that if we take the goal here was to obtain a gravity solution “tuned” to LRO, average performance over the nominal mission only, the orbit with the orbital performance as metric. quality is further improved to ∼15 m RMS. While this could Nevertheless, the addition of the LRO dataset can poten- be due to the specifics of the later phases (SM_01 to SM_05), tially benefit the estimation of low-degree terms in the gravi- it is plausible that those latter phases would see additional tational signal, given the longer arc duration (5 days) and their improvements with a gravity solution that includes data from links, through crossovers, over monthly timescales. Table 2 those phases. In other words, the overlap numbers probably lists our estimates in comparison to previous results (Dickey reflect orbit reconstruction performance for the earlier phases et al. 1994; Williams et al. 2009, 2011; Konopliv et al. and something closer to orbit prediction performance for the 2001; Goosens and Matsumoto 2008; Matsumoto et al. 2010; latter ones. This will be answered by continuing work and Goossens et al. 2011; Mazarico et al. 2010). The LLGM-1 iterations on the LRO gravity fields. adjusted values are generally consistent with the existing With the inclusion of the altimetric data in the orbit solutions, in particular LP100K and LP150Q (Konopliv et al. convergence, similar observations can be made, but the orbit 2000, 2001), except for C21 and k2.OurC21 estimate is rather consistency is strengthened even further (Table S4). The large and of negative sign, compared to previous estimates calculated orbit overlap RMS over the whole mission are that were generally small and positive. Although it is not ∼8.3, 8.2 and 1.6 m in the along-track, cross-track and radial correlated with S21 (−0.07), the covariance matrix shows directions, respectively, and 12.0m in total position. The rela- that the correlation coefficients of C21 with C30, C32 and tive improvement is much smaller than the 70% obtained with 123 E. Mazarico et al.

abnavigation orbits LLGM-1 orbits 30 30

km 1.5 20 20

0.5 10 10

-0.5 0 0

-1.5 -10 -10

-2.5 -20 -20

-3.5 -30 -30 -30 -20 -10 0 10 20 30 -30 -20 -10 01020 30

Fig. 6 Topographic maps in polar stereographic projection of the lunar north pole region (axis are in kilometers), created from LOLA data geolocated with the navigation orbits (a) and with the LLGM-1 orbits (b). The map resolution is 20 m (true scale at the pole)

GLGM-3, but still significant (19%) given the magnitude of measure of precision and not of the orbit accuracy itself the radiometric-only errors. The gain in the radial direction (as required for the LRO position knowledge; Vondrak et al. is larger (about 50%), but does not yield significantly smaller 2010). The orbit accuracy is difﬁcult to assess with the track- crossover discrepancies (Table 1). We note that on average ing measurement types we used, because the true trajectory is the radial direction is slightly degraded by the addition of the of course unknown. Here, we use independent data to show altimetric crossovers. However, the crossover discrepancies that the orbit precision numbers (Sects. 3.2, 3.4) are com- are now commensurate with the radial overlap RMS values. mensurate with the orbit accuracy. The crossover residuals are smaller near the poles, where the Although we used a small amount of LOLA data in the majority of crossovers occur. The GLGM-3 overlaps were POD process to construct the altimetric crossovers, the over- too large to observe this effect, but it was expected from the whelming majority of the altimetric measurements (laser Rowlands et al. (2009) LRO simulation study: the crossovers bounce points) did not directly contribute to the orbit adjust- tend to distribute the orbital errors among all directions, cor- ment. The topographic maps produced from the complete rupting the empirical rule of the radial errors being ten times LOLA dataset can thus be viewed as a legitimate test of the smaller than the horizontal errors. That rule still holds rela- orbit accuracy, especially in the polar regions where the data tively well for the radiometric-only (both with GLGM-3 and density is high. In polar stereographic projection, with the LLGM-1), leading to larger radial errors when including the orbits rotating with a monthly period, the sharpness of the LOLA data. We ﬁnd that the orbit differences between the topographic maps is possible only with accurate orbits. radiometric-only orbits and those that include the crossovers In Fig. 6, we compare maps of the lunar north polar region (10.8, 7.3, 1.5 and 14.2 m on average in the ACR directions (89–90◦ N) at 20-m resolution created with the navigation and total position, respectively) are commensurate with the orbits (Fig. 6a) and with the orbits presented in Sect. 3.4 differences in their overlap RMS values (Table 3), which (Fig. 6b). Although artifacts due to orbit errors are still pres- is consistent with the crossovers bringing largely favorable ent in Fig. 6b, they are significantly reduced and are the result orbital changes. of a small number of tracks. Overall, the geolocation is excellent and broadly consistent with the overlap RMS decrease compared to other solutions (Table 3), and with the expected 4 Discussion and closing remarks 20-m precision level determined from overlaps. Images acquired with the LROC NAC camera, at 50-cm 4.1 Orbit accuracy assessment from independent data resolution, can also provide strong measures of the orbit accuracy. The LROC instrument is repeatedly pointed at the The overlap analysis above shows that the LRO orbits are Apollo landing sites, as Constellation sites, but also as opti- consistent at the 10- to 15-m level. This is technically a mal calibration targets. To calibrate the instrument pointing 123 LRO orbit determination

Table 4 Summary of the statistics of the timing residuals Target Number of Timing errors (ms) Timing errors (ms) within LROC NAC images of observations navigation orbits LLGM-1 orbits the predicted and observed positions of Apollo/Lunokhod Mean RMS Mean RMS artifacts Apollo 11 10 −7.19 11.12 −2.83 6.19 Apollo 12 11 −9.98 28.58 0.44 6.13 Apollo 14 13 −6.36 12.31 −0.54 12.36 Apollo 15 11 −4.75 7.18 −5.38 6.56 Apollo 16 10 −5.16 19.61 −4.62 12.84 Apollo 17 11 −18.46 36.54 −4.40 8.96 Lunokhod 1 7 −20.52 35.21 −2.98 9.3 Results with both the navigation Average 10.4 −10.35 21.51 −2.90 8.91 and the LLGM-1 orbit Weighted average −−9.79 20.67 −2.81 8.96 reconstructions are given

and timing, the LROC team has been keeping track of the and near 1 m radially. A gravity ﬁeld solution that combined position of known human artifacts in the LROC NAC image pre-LRO tracking data with the newly acquired LRO geodetic frames. Those data (M.S. Robinson, personal communica- measurements was developed and tuned to further improve tion) show that the orbits obtained in this work achieve the LRO trajectory. This resulted in accuracies better than improved accuracy, reducing the scatter of timing residu- 20 m over the whole mission (July 2009 to January 2011). als (between expected and actual) from ∼21 ms RMS to This work will be continued to provide a consistent accuracy 9 ms RMS (Table 4, Figure S6). The mean of the residuals of the trajectory for all the data acquired by LRO and to estab- decreases significantly as well, from about −10 to −3ms. lish a high-quality lunar reference frame to combine datasets Given the ∼1.7kms−1 LRO orbital velocity, those numbers of multiple spacecraft, past or future. Further enhancements are in good agreement with the results of the overlap analysis can also be expected: by the creation of multi-month arcs to (i.e., ∼15 m RMS, with an average of ∼5m). make use of a larger number of altimetric crossovers; and ultimately by a reprocessing of the data with a much-improved gravity ﬁeld solution obtained by the GRAIL mission (Zuber 4.2 Creation of products and distribution et al. 2011).

The orbits presented in Sect. 3.4 are made available as part Acknowledgments We thank M.S. Robinson and the LROC Science of the LRO Radio Science PDS archive (http://imbrium.mit. Team for providing the data used to independently validate the position- edu/LRORS/). They are distributed as a set of SPICE ker- ing accuracy. D. Pavlis (SGT Inc.) and C. Deng (Innovim Inc.) provided nels (Acton 1996), each spanning one of the monthly phases assistance with the modeling of the LRO crossover observations in GE- ODYN. We thank the LRO project and the ﬂight team for the timely described in Sect. 2.1. A linear taper is used in the overlap delivery of mission products and related information. We thank three periods to combine the trajectories of consecutive arcs. The anonymous reviewers for their thoughtful comments. LLGM-1 gravity ﬁeld is also made available through the PDS archive or by contacting the authors. References

4.3 Summary of results Acton CH (1996) Ancillary data services of NASA’s navigation and ancillary information facility. Planet Space Sci 44:65–70. doi:10. 1016/0032-0633(95)00107-7 In this paper, we presented the data and methods used by Araki H, Tazawa S, Noda H, Ishihara Y, Goossens S, Sasaki S, the LOLA POD team to reconstruct the LRO spacecraft tra- Kawano N, Kamiya I, Otake H, Oberst J, Shum C (2009) Lunar jectory to satisfy the LRO position knowledge requirements global shape and polar topography derived from Kaguya-LALT and to enable the geolocation and comparison of the various laser altimetry. Science 323(5916):897–900. doi:10.1126/science. 1164146 high-resolution datasets being acquired by the instruments Chin G, Brylow S, Foote M, Garvin J, Kasper J, Keller J, Litvak M, onboard the spacecraft. We demonstrated through the use of Mitrofanov M, Paige D, Raney K, Robinson M, Sanin A, Smith D, overlap analysis and with the help of additional independent Spence H, Spudis P, Stern SA, Zuber M (2007) Lunar Reconnais- datasets that the orbits obtained with radiometric tracking sance Orbiter overview: the instrument suite and mission. Space Sci Rev. doi:10.1007/s11214-007-9153-y data supplemented by LOLA-derived altimetric constraints Dickey JO, Bender PL, Faller JE, Newhall XX, Ricklefs RL, Ries achieve position accuracy levels around 20 m in total position JG, Shelus PJ, Veillet C, Whipple AL, Wiant JR, Williams JG, 123 E. Mazarico et al.

Yoder CF (1994) Lunar laser ranging: a continuing legacy of the RF) technology demonstration. Space Sci Rev 150:285–302. Apollo program. Science 265:482–490. doi:10.1126/science.265. doi:10.1007/s11214-009-9607-5 5171.482 Paige DA, Foote MC, Greenhagen BT, Schofield JT, Calcutt S, Vasa- Floberghagen R, Visser P, WeischedeF (1999) Lunar albedo force mod- vada AR, Preston DJ, Taylor FW, Allen CC, Snook KJ, Jakosky eling and its effect on low lunar orbit and gravity field deter- BM, Murray BC, Soderblom LA, Jau B, Loring S, Bulharowski J, mination. Adv Space Res 23(4):733–738. doi:10.1016/S0273- Bowles NE, Thomas IR, Sullivan MT, Avis C, De Jong EM, Hart- 1177(99)00155-6 ford W, McClees DJ (2010) The Lunar Reconnaissance Orbiter Goosens S, Matsumoto K (2008) Lunar degree 2 potential Love num- diviner lunar radiometer experiment. Space Sci Rev 150:125–160. ber determination from satellite tracking data. Geophys Res Lett doi:10.1007/s11214-009-9529-2 35:L02204. doi:10.1029/2007GL031960 Pavlis DE, Poulose SG, McCarthy JJ (2006) GEODYN operations man- Goossens S, Matsumoto K, Liu Q, Kikushi F,Sato K, Hanada H, Ishihara uals. Contractor Report, SGT Inc., Greenbelt, Maryland Y,Noda H, Kawano N, Namiki N, Iwata T, Lemoine FG, Rowlands Robinson MS, Brylow SM, Tschimmel M, Humm D, Lawrence SJ, DD, Harada Y, Chen M (2011) Lunar gravity field determination Thomas PC, Denevi BW, Bowman-Cisneros E, Zerr J, Ravine MA, using SELENE same-beam differential VLBI tracking data. J Geod Caplinger MA, Ghaemi FT, Schaffner JA, Malin MC, Mahanti P, 85(4):205–228. doi:10.1007/s00190-010-0430-2 Bartels A, Anderson J, Tran TN, Eliason EM, McEwen AS, Turtle Houghton MB, Tooley CR, Saylor RS Jr (2007) Mission design E, Jolliff BL, Hiesinger H (2010) Lunar Reconnaissance Orbiter and operations considerations for NASA’s Lunar Reconnaissance Camera (LROC) instrument overview. Space Sci Rev 150:81–124. Orbiter. 58th International Astronautical Congress, Hyderabad, doi:10.1007/s11214-010-9634-2 India. http://lunar.gsfc.nasa.gov/library/IAC-07-C1_7_06.pdf Rowlands DD, Pavlis DE, Lemoine FG, Neumann GA, Lutchke Kaula WM (1966) Theory of satellite geodesy. Blaisdell Publishing SB (1999) The use of laser altimetry in the orbit and attitude deter- Co., London 124 mination of Mars Global Surveyor. Geophys Res Lett 26:1191– Konopliv AS, The Lunar Prospector Gravity Science Team (2000) 1194. doi:10.1029/1999GL900223 LP150Q Spherical Harmonic Model. http://pds-geosciences. Rowlands DD, Lemoine FG, Chinn DS, Luthcke SB (2009) A simula- wustl.edu/lunar01/lp-l-rss-5-gravity-v1/lp_1001/sha/jgl150q1. tion study of multi-beam altimetry for lunar reconnaissance orbiter sha (published 27 Nov. 2000) and other planetary missions. J Geod 83(8):709–721. doi:10.1007/ Konopliv AS, Asmar SW, Carranza E, Sjogren WL, Yuan D- s00190-008-0285-y N (2001) Recent gravity models as a result of the Lunar Prospector Smith DE, Zuber MT, Frey HV, Garvin JB, Head JW, Muhleman DO, mission. Icarus 150:1–18. doi:10.1006/icar.2000.6573 Pettengill GH, Phillips RJ, Solomon SC, Zwally HJ, Banerdt WB, Lemoine FG, Smith DE, Zuber MT, Neumann GA, Rowlands Duxbury TC, Golombek MP, Lemoine FG, Neumann GA, Row- DD (1997) GLGM-2: a 70th degree and order Lunar Gravity lands DD, Aharonson O, Ford PG, Ivanov AB, McGovern PJ, Model from Clementine and historical data. J Geophys Res Abshire JB, Afzal RS, Sun X (2001) Mars Orbiter Laser Altim- 102(E7):16339–16359. doi:10.1029/97JE01418 eter (MOLA): experiment summary after the first year of global Lemoine FG, Smith DE, Rowlands DD, Zuber MT, Neumann GA, mapping of Mars. J Geophys Res 106:23689–23722. doi:10.1029/ Chinn DS, Pavlis DE (2001) An improved solution of the gravity 2000JE001364 field of Mars (GMM-2B) from Mars Global Surveyor. J Geophys Smith DE, Zuber MT, Jackson GB, Cavanaugh JF, Neumann GA, Riris Res 106:23359–23376. doi:10.1029/2000JE001426 H, Sun X, Zellar RS, Coltharp C, Connelly J, Katz RB, Kleyner Luthcke SB, Zelensky NP, Rowlands DD, Lemoine FG, Williams I, Liiva P, Matuszeski A, Mazarico E, McGarry J, Novo-Gradac TA (2003) The 1-centimeter orbit: Jason-1 precise orbit determi- A-M, Ott MN, Peters C, Ramos-Izquierdo LA, Ramsey L, Row- nation using GPS, SLR, DORIS and altimeter data. Mar Geod lands DD, Schmidt S, Scott VS, Shaw GB, Smith JC, Swinski 26(3-4):399–421. doi:10.1080/714044529 J-P, Torrence MH, Unger G, Yu AW, Zagwodzki TW (2009) The Matsumoto K, Goossens S, Ishihara Y, Liu Q, Kikuchi F, Iwata T, Nam- Lunar Orbiter Laser Altimeter investigation on the Lunar Recon- iki N, Noda H, Hanada H, Kawano N, Lemoine FG, Rowlands naissance Orbiter mission. Space Sci Rev 150:209–241. doi:10. DD (2010) An improved lunar gravity field model from SELENE 1007/s11214-009-9512-y and historical tracking data: revealing the farside gravity features. Smith DE, Zuber MT, Neumann GA, Lemoine FG, Mazarico E, Tor- J Geophys Res 115:E06007. doi:10.1029/2009JE003499 rence MH, McGarry JF, Rowlands DD, Head JW, Duxbury TH, Mazarico E, Zuber MT, Lemoine FG, Smith DE (2009) Effects of Aharonson O, Lucey PG, Robinson MS, Barnouin OS, Cavanaugh self-shadowing on nonconservative force modeling for mars-orbit- JF, Sun X, Liiva P, Mao D-D, Smith JC, Bartels AE (2010) Ini- ing spacecraft. J Spacecr Rockets 46(3):662–669. doi:10.2514/1. tial observations from the Lunar Orbiter Laser Altimeter (LOLA). 41679 Geophys Res Lett 37:L18204. doi:10.1029/2010GL043751 Mazarico E, Lemoine FG, Han S-C, Smith DE (2010) GLGM-3: a Smith DE, Zuber MT, Phillips RJ, Solomon SC, Neumann GA, Lemo- degree-150 lunar gravity model from the historical tracking data of ine FG, Peale SJ, Margot J-L, Torrence MH, Talpe MJ, Head JW, NASA Moon orbiters. J Geophys Res 115:E05001. doi:10.1029/ Hauck SA, Johnson CL, Perry ME, Barnouin OS, McNutt RL, 2009JE003472 Oberst J (2010) The equatorial shape and gravity field of Mercury Namiki N, Iwata T, Matsumoto K, Hanada H, Noda H, Goossens S, from MESSENGER flybys 1 and 2. Icarus 209(1):88–100. doi:10. Ogawa M, Kawano N, Asari K, Tsuruta S, Ishihara Y, Liu Q, Ki- 1016/j.icarus.2010.04.007 kuchi F, Ishikawa T, Sasaki S, Aoshima C, Kurosawa K, Sugita S, Vondrak R, Keller J, Chin G, Garvin J (2010) Lunar Reconnaissance Takano T (2009) Farside gravity field of the Moon from four-way Orbiter (LRO): observations for lunar exploration and science. Doppler measurements of SELENE (Kaguya). Science 323:900– Space Sci Rev 150:7–22. doi:10.1007/s11214-010-9631-5 905. doi:10.1126/science.1168029 Williams JG, Newhall XX, Dickey JO (1987) Lunar gravitational har- Neumann GA, Mazarico E, Smith DE, Zuber MT, Gläser P (2011) Lunar monics and reflector coordinates. In: Holota P (ed) Proceedings of Orbiter Laser Altimeter measures of slope and roughness. Lunar the international symposium on figure and dynamics of the Earth, and Planetary Science Conference XXXXII. The Woodlands, TX Moon and Planets. Astronomical Institute of the Czechoslovak (abstract 2313) Academy of Sciences, Prague, pp 643–648 Nozette S, Spudis P, Bussey B, Jensen R, Raney K, Winters H, Lichten- Williams JG, Boggs DH, Folkner WM (2008) DE421 Lunar Orbit, berg CL, Marinelli W, Crusan J, Gates M, Robinson M (2010) The Physical Librations, and Surface Coordinates. JPL Memorandum Lunar Reconnaissance Orbiter miniature radio frequency (Mini- IOM 335-JW,DB,WF-20080314-001. March 14, ftp://ssd.jpl. 123 LRO orbit determination

nasa.gov/pub/eph/planets/ioms/de421_moon_coord_iom.pdf, Zuber MT, Smith DE, Zellar RS, Neumann GA, Sun X, Katz RB, Kley- ftp://naif.jpl.nasa.gov/pub/naif/generic_kernels/spk/planets/ ner I, Matuszeski A, McGarry JF, Ott MN, Ramos-Izquierdo LA, de421_lunar_ephemeris_and_orientation.pdf Rowlands DD, Torrence MH, Zagwodzki TW (2010) The Lunar Williams JG, Boggs DH, Ratcliff JT (2009) A larger lunar core? In: Reconnaissance Orbiter laser ranging investigation. Space Sci Rev Lunar and Planetary Science Conference XXXX, The Woodlands, 150:63–80. doi:10.1007/s11214-009-9511-z TX (abstract 1452) Zuber MT, Smith DE, Watkins MM, Lehman DH, Hoffman TL, Asmar Williams JG, Boggs DH, Ratcliff JT (2011) Lunar moment of iner- SW, Konopliv AS, Lemoine FG, Melosh HJK, Neumann GA, Phil- tia and love number. In: Lunar and Planetary Science Conference lips RJ, Solomon SC, Wieczorek MA, Williams JG (2011) Gravity XXXXII, The Woodlands, TX, (abstract 2610) Recovery and Interior Laboratory (GRAIL) mission and science Zuber MT, Smith DE, Solomon SC, Muhleman DO, Head JW, Gar- objectives. Space Sci Rev (submitted) vin JB, Abshire JB, Bufton JL (1992) The Mars Observer Laser Altimeter investigation. J Geophys Res 97:7781–7797

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